How Does the Volume Change?

Date: 03/04/99 at 10:05:59
From: Caleb
Subject: How does the volume change when you change one side?
Recently in math class, we have been doing a project where we change
one side of an object and see how it affects the volume. If you are
doing this to a triangular prism and your formula is 1/2 the base
times length times height, could you cut the 1/2 in half instead of
the base, length, or height and get the same results?

Date: 03/04/99 at 13:06:10
From: Doctor Rick
Subject: Re: How does the volume change when you change one side?
It sounds as if you have noticed that whatever numbers you put in the
formula, when you make the base, length or height half as big, the
answer is the same as
1/4 * base * length * height
and you want to know if this will always work.
If so, the answer is yes. What you are asking about is algebra. If you
have a triangular prism and you reduce the length (for example) by a
factor of 2, the volume of the new prism is
Vnew = 1/2 base * (length/2) * height
You can rearrange the operations using the rules of algebra (the
commutative and associative principles, in particular). It does not
matter what numbers base, length, and height stand for; these will
always be equal:
1/2 * base * (length/2) * height
1/2 * base * (1/2 * length) * height (commutative principle)
1/2 * (base * 1/2) * length * height (associative principle)
1/2 * (1/2 * base) * length * height (commutative principle again)
(1/2 * 1/2) * base * length * height (associative principle again)
1/4 * base * length * height
This formula will work for any base, length, and height, because the
commutative and associative principles are true for any numbers.
If this is new to you, then you are discovering for yourself how
algebra works, and that is great. If you have seen it before, I hope
this helps you see how these principles can be put to work for you.
- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/